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1 A Structure for General and Specc Market Rsk Eckhard Platen 1 and Gerhard Stahl Summary. The paper presents a consstent approach to the modelng of general and specc market rsk as dened n regulatory documents. It compares the statstcally based betafactor model wth a class of benchmark models that use a broadly based ndex as major buldng block for modelng. The nvestgaton of logreturn of stock prces that are expressed n unts of the market ndex reveals that these are lkely to be Student t dstrbuted. A correspondng dscrete tme benchmark model s then used to calculate ValueatRsk for equty portfolos. Key words: Rsk measurement, general market rsk, specc market rsk, Value at Rsk, nancal modelng, benchmark model, growth optmal portfolo. 1 Introducton Tradng portfolos of nancal nsttutons are characterzed by nonlnear nstruments, ted to complex tradng strateges. The nomnal volume of such postons s n general not proportonal to the rsk that s taken. Fnancal nsttutons can run nternal models for calculatng regulatory captal, see Basle (1996a, 1996b). These models provde forecast dstrbutons of portfolo losses due to uctuatons of market prces. In ths context t s mportant to see how regulatory terms are translated nto quanttatve rsk modelng. Market rsk, whch s due to uctuatons of market prces, plays an essental role n determnng regulatory captal. It s understood as the core rsk that an nsttuton s exposed to through ts tradng portfolo. Market rsk s splt nto general and specc market rsk. More precsely, for an equty portfolo general market rsk denotes the rsk exposure of the portfolo aganst the equty market as a whole. On the other hand, specc market rsk relates to the rsk of holdng an ndvdual securty wthn an equty portfolo, whch s not covered by general market rsk. Specc market rsk can be decomposed 1 Unversty oftechnology Sydney, School of Fnance & Economcs and Department of Mathematcal Scences, PO Box 13, Broadway, NSW, 007, Australa German Fnancal Supervsory Authorty, Graurhendorferstr. 108, D Bonn. The vews n ths paper should not be construed as endorsed by the GFSA.
2 nto dosyncratc and event rsk. Ths dstncton s used because events lke mergers, earnngs surprses, bankruptces and ratng mgratons are key nputs for the securty dynamcs. The separaton of market rsk nto ts general and specc components has sgncant mpact on the amount of regulatory captal requred to cover the market rsk of a tradng book. In the framework of nternal models ths captal charge s determned by means of a rsk measure, the ValueatRsk (VaR). Ths paper addresses ssues arsng from the applcaton of the current regulatory approach. The rch lterature on VaR comprses, for nstance, RskMetrcs (1996), Alexander (1996), Joron (000), Due & Pan (1997, 001) and Embrechts, McNeal & Straumann (1999). A sutable metrc for rsk measurement s deally derved by an adequate parsmonous modelng structure that ncorporates the essental sources of uncertanty. As prces are relatve, such a structure should dene an approprate reference unt to be used as numerare or benchmark n establshng a correspondng metrc for measurng rsk. In ths paper we suggest a benchmark approach, where we consstently use a broadly based ndex (BBI) or market wde ndex as benchmark. Ths denes a natural coordnate system n our market whch goesbeyond the regresson based betafactor model and consderably mproves the measurement of market rsk. Furthermore, as shown n Platen (00), a BBI approxmates under general condtons the growth optmal portfolo (GOP), see Kelly (1956). Usng the GOP as reference unt the resultng benchmark model has a number of nterestng and useful propertes, see Platen (001a, 001b). We analyze n ths paper logreturns of equty prces when these are expressed n unts of the equty market ndex. Strong evdence s shown that these are Student t dstrbuted wth degrees of freedom that range typcally between 3 and 5. Ths leads to the speccaton of a Student tbenchmark model that can be shown to yeld VaR numbers consstent wth emprcal ndngs. Dscrete Tme Market Let us consder a dscrete tme equty and xed ncome market. Prces are assumed to change ther values only at the gven dscrete, equdstant tmes 0 t 0 <t 1 <:::<t n < 1 for n f0 1 :::g. The tme step sze s denoted by =t +1 ; t whch s typcally one day, f0 1 ::: n; 1g. We consder d +1 prmary assets, d f1 :::g and denote by S the value at tme t of the jth prmary securty account, whchstypcally an equty or bond wth all dvdends and coupon payments renvested. We assume throughout the paper that S > 0 (.1)
3 for all j f0 1 ::: dg and f0 1 ::: ng. We assume that S (0) s the rskless domestc savngs account at tme t, whch s a rollover short term bond account. The return R +1 of the jth prmary securty account at tme t +1 s dened as R +1 = S +1 ; S S (.) for f0 1 ::: n;1g and j f0 1 ::: dg. The noton of a return s smple and well establshed. Equvalently, we also ntroduce the growth rato H +1 of the jth prmary securty account attmet +1 n the form H +1 = R +1 +1= S +1 S (.3) for f0 1 ::: n; 1g and j f0 1 ::: dg. Ths leads us naturally to the ntroducton of the jth logreturn at tme t +1 n the form L +1 =log H +1 : (.4) Note, for typcal daly prce movements the return R +1 approxmates well the logreturn L +1. For the characterzaton of a portfolo at tme t t s sucent to descrbe the vector of proportons = ( (1) 1 ::: (d) ) >, wth (;1 1) denotng the proporton of the value of the portfolo at tme t that s nvested n the jth prmary securty account, j f0 1 ::: dg. By a > we denote the transpose of a vector or a matrx. Obvously, the proportons sum to one, that s dx j=0 =1 (.5) for all f0 1 ::: ng. The value of the correspondng portfolo at tme t s denoted by S (), f0 1 ::: ng. In ths context the number j of unts of the jth prmary securty account attmet s gven by the relaton j = S () S (.6) for j f0 1 ::: dg and f0 1 ::: ng. We obtanthegrowth rato H () ` of ths portfolo at tme t` n the form H () ` = dx j=0 `;1 H ` (.7) 3
4 whch, smlarly to (.3), can be wrtten as the portfolo return for ` f1 ::: ng. R () ` = H () ` ; 1= dx j=0 `;1 R ` (.8) 3 Regulatory Termnology and Framework Before we consder specc regulatory terms, we recall the denton of VaR. We denote by VaR h (S () ) the VaR number at tme t of a gven portfolo S () wth proportons, a gven level of sgncance and a forecast horzon of h tradng days. In practce, h s typcally chosen as one or ten tradng days, that s h f1 10g. More precsely, the VaR number VaR h (S () ) denotes the quantle of the dstrbuton functon of the random varable S () h = S() ; ~ S () +h (3.1) where S () s known at tme t. The varable S ~ () denotes at tme +h t +h the random value at the tme t xed portfolo, that s ~S () +h = dx j=0 j S +h where the number of unts j to be held n the jth prmary securty account at tme t s gven n (.6). Typcally, the sgncance level s set to 99%. The VaR number s nterpreted as an upper bound of losses, expressed n unts of the domestc currency that mght only be surpassed wth probablty 1 ;. These losses are caused by prce changes of the underlyng securtes n the portfolo. Next we ntroduce the ocal denton of market rsk, see Basle (1996a), p.1: Market rsk s dened as the rsk of losses n on and obalancesheet postons arsng from movements n market prces. The rsks subject to ths requrement are: the rsks pertanng to nterest rate related nstruments and equtes n the tradng book foregn exchange rsk and commodty rsk throughout the bank. As outlned n the ntroducton, market rsk can be dvded nto general and specc market rsk. The followng denton of general market rsk s gven n Basle (1996a), p. 19 and p. 9: General market rsk covers the rsk of holdng long or short postons n nterest rate or equty rsk aganst the market as a whole. In Basle (1996a), p. ths s made more precse: The market should be dented wth a sngle factor that s representatve for the market as a whole, for example, a wdely accepted broadly based stock ndex for the country concerned. 4
5 We emphasze, t s a regulatory requrement that a broadly based ndex (BBI) serves as a reference unt. We wll naturally ncorporate ths requrement n our approach by usng a BBI as benchmark, denoted by S (). The followng denton relates specc rsk, whch ncludes event rsk, to ndces, see Basle (1996a), p. 5: Specc rsk ncludes the rsk that an ndvdual debt or equty securty moves by more orless than the general market n daytoday tradng, ncludng perods when the whole market s volatle. Specc rsk covers that rsk n holdng long or short postons n an ndvdual equty or debt securty. Event rsk covers the rsk, where the prce of an ndvdual debt or equty securty moves precptously relatve to the general market due to a major event, e.g., on a takeover bd or some other shock event such events would also nclude the rsk of default. The derentaton between derent forms of rsk allows a bank to talor regulatory captal, that s captal cushons or reserves that correspond to the nherent rsks of a gven portfolo, see Basle (1996a), p.. Assume that an nternal model provdes for a portfolo S (), based on the regulatory parameters h = 10 and = 99%, the VaR number VaR 10 (S () 99%) at tme t for general market rsk. Furthermore, suppose that a prescrpton s gven to calculate separately the specc rsk wth the assocated VaR gure denoted by VaR S 10 (S() tme t for the portfolo S () C R = max 99%). In order to determne the regulatory captal C R at the followng formula has to be appled: VaR 10 S () 99% + m VaR S 10 S () 99% M 1 60 X59 `=0 VaR 10 S () ;` 99% + m 1 60 X59 l=0 VaR S 10 S () ;` 99% : Internal models that cover dosyncratc rsk but not event rsk are called surcharge models. In that case the varable m equals 1. For those models that cover specc rsk ncludng event rsk,m s set to zero. M denotes a safety multpler whch s usually set to 3. We remark, that event rsk, when t s covered by general market rsk, does not requre partcular regulatory captal. For such rsks that are related to low probablty or rare events, socalled, stress tests have to be appled, see Basle (1996a), p. 46 and Gbson (001). 4 Beta Factor Model It s common practce, see RskMetrcs (1996) and Basle (1996a), to regress the return R +1 of the jth prmary securty account at tme t +1 on the, so called, jth beta factor to separate the mpact of general and specc market rsk. To apply a beta factor model for equtes, a system of lnear 5
6 regresson equatons s then used, where R +1 = R(0) +1 + R () +1 + " +1 (4.1) for f0 1 ::: n ; 1g and j f0 1 ::: dg. Recall that R (0) +1 return of the domestc savngs account and R () +1 s the s that of the BBI. Let E denote condtonal expectaton gven the nformaton up untl tme t. Furthermore, n a beta factor model the Gaussan random varable " the jth dosyncratc nose and R () +1 are assumed to be such that E " +1 E " +1 E " +1 "(`) +1 = 0 E " +1 R() +1 = E R () " +1 = 0 E R +1 R() +1 =0 = = () () + E + R (0) +1 E R () for R () +1 for f0 1 ::: n; 1g and j ` f0 1 ::: dg wth ` 6= j. Note that the return R +1 of the jth prmary securty account n (4.1) depends lnearly on the return R () +1 of the BBI S(). The jth dosyncratc nose " +1, s nether correlated to the market return R () +1 nor to the other dosyncratc nose terms " (k) +1 for k 6= j. As a result of these assumptons, the varance of the return of the jth prmary securty account can be decomposed nto the sum = E R +1 ; E R +1 =( ) () + (4.) " (4.3) for f0 1 ::: n; 1g and j f0 1 ::: dg, wththevarance of the return of the BBI () = E R () +1 ; E R +1 () : (4.4) In ths setup the returns and also ther varances, see (4.1) and (4.3), are lnearly related. Equaton (4.3) expresses the total market rsk of the jth prmary securty account n the form of varances. The rst and second term express the general and specc market rsk, respectvely. The above regresson based beta factor model s n accordance wth the regulatory dentons gven n Secton 3. The quantty R () +1 the socalled betaequvalent of the return R +1 n (4.1) s. Note that relaton (4.3) s 6
7 smply a lnear decomposton of market rsk nto general and specc market rsk on the level of second moments. The beta factor model, whch s a lnear regresson model wth constant beta factor, has a purely statstcal motvaton. Addtonally, both the lnearty and Gaussan assumptons mposed on returns provde a rather crude approxmaton to observed returns. In realty, returns exhbt much fatter tals n ther dstrbutons as we wll see below. 5 Benchmark Framework In the followng, we consder an alternatve to the beta factor model. The benchmark framework allows us to separate general and specc market rsk n a canoncal way. Ths separaton s acheved n a natural settng usng a BBI S () as reference unt or benchmark. We ntroduce the jth benchmarked growth rato ^H +1 H () = H (5.1) for f0 1 ::: n; 1g and j f0 1 ::: dg, see (.3) and (.7). Equaton (5.1) allows us to express the growth rato of the jth prmary securty account as the product H +1 = H () +1 ^H +1 (5.) for f0 1 ::: n ; 1g and j f0 1 ::: dg. Ths product provdes a multplcatve decomposton of the jth growth rato. The rst factor H () ^H s related to general market rsk, whereas the second factor s naturally ted to the specc market rsk of the jth prmary securty account at tme t +1. In ths benchmark framework we denote the logarthm of the jth benchmarked growth rato (5.1), by ^L +1 =log ^H +1 (5.3) for f0 1 ::: n; 1g and j f0 1 ::: dg. Recall that we take the tme step sze to be small. We thus obtan the condtonal expectaton E ^L +1 (5.4) whch can be assumed to be small wth some nte random varable. Note that these knd of approxmatons can be made precse n a contnuous tme settng where we let the tme step sze tend to zero. Comparng the returns of normalzed BBIs, for nstance, the S&P100, S&P500, S&P1000 and the MSCI world ndex, t s clear that these ndces behave n a very smlar manner. In Platen (00) t has been shown that 7
8 the movements of such potental BBIs approxmate those of the GOP. Thus the GOP or a proxy arses naturally when modelng general market rsk by a BBI. 6 Semparametrc Benchmark Models Let us ntroduce a general class of semparametrc benchmark models, where we assume that the jth centralzed logreturn X +1 = ^L ^L +1 ; E +1 (6.1) admts the structure X +1 = ; dx k=1 p j k (k) Z +1 (6.) for j f0 1 ::: dg and f0 1 ::: n; 1g. Note that X (0) +1 s the centralzed logreturn of the benchmarked domestc savngs account. Here, j k s called the jth volatlty at tme t wth respect to the kth source ofuncertanty Z (k) +1. We choose Z(1) +1 ::: Z(d) +1 as random varables wth zero mean E Z (k) +1 =0 (6.3) unt varance and such that E Z (k) +1 =1 (6.4) E Z (k) Z(`) =0 (6.5) for ` 6= k wth f0 1 ::: n; 1g and k ` f1 ::: dg. We assume for techncal reasons that an absolute moment of order slghtly greater than two exst for the vector of uncertanty Z +1 =(Z (1) +1 ::: Z(d) +1 )>. Note that n contrast to the beta factor model, we are not restrcted to the use of Gaussan random varables. Nor do we assume the ndependence of market returns and benchmarked ndvdual returns. We obtan from (6.)  (6.5) the second order normalzed condtonal moments c j ` = 1 E X +1 X (`) +1 = dx k=1 j k ` k (6.6) 8
9 for f0 1 ::: n; 1g and j ` f1 ::: dg. Relaton (6.6) allows us to ntroduce the covarance matrx as the product wth volatlty matrx =[c j ` ] d j `=1 (6.7) = D D > (6.8) D =[ j ` ] d j l=1 (6.9) f0 1 ::: n; 1g. The volatlty matrx D can be nterpreted as the Cholesky decomposton of. If the volatlty matrx D s nvertble, then, by (6.), the vector Z +1 =(Z (1) +1 ::: Z(d) +1 )> of the sources of uncertanty can be explctly expressed n the form ; 1 p D ;1 X +1 = Z +1 (6.10) usng the vector of observed growth ratos X +1 = (X (1) +1 ::: X(d) +1 )>, f0 1 ::: n; 1g. By equatons (5.1) and (6.1) wth j = 0 t can be seen that X (0) +1 = = log (0) ^L ^L(0) +1 ; E +1 H (0) +1 ; E log H (0) +1 ; log H () +1 + E log H () +1 for f0 1 ::: n; 1g. Snce the growth rato H (0) +1 of the savngs account s known at tme t the rst two terms n the above formula oset each other. Thus, we obtan from the fact that E (log(h () +1 )) () than p, the logreturn of the BBI n the approxmate form log H () +1 = ;X (0) +1 + E log H () +1 s of hgher order ;X (0) +1 (6.11) for f0 1 ::: n; 1g. Ths means, the uncertanty of the logreturn of the BBI s approxmately equal to the negatve of that of the benchmarked savngs account. By assumng the return R +1 of the jth prmary securty account tobe small we obtan from relaton (.3) and the expanson of the logarthm that R +1 = H +1 ; 1 log H +1 (6.1) 9
10 for j f0 1 ::: dg. Now, relaton (5.) yelds R +1 log H +1 =log H () +1 +log ^H +1 and t follows from (6.11), (5.3) and (6.1) by neglectng hgher order terms that R (0) +1 ;X +1 + X +1 : (6.13) The above descrbed semparametrc benchmark model s based on the multplcatve relatonshp (5.) between growth ratos. These naturally express general and specc market rsk. As s evdent from (6.11) and(6.), provdes a measure of the exposure of the BBI towards the kth source of uncertanty Z (k) +1. Smlarly, by (5.3), (6.1) and (6.) the volatlty j k quantes the exposure of the jth benchmarked prmary securty account towards the kth source of uncertanty. One can say, the volatltes 0 k of the BBI parameterze the general market rsk, whereas the volatlty j k reects the specc market rsk of the jth prmary securty account wth respect to the kth source of uncertanty, f0 1 ::: n; 1g, j f0 1 ::: dg and k f1 ::: dg. Let us now provde a lnk to the betafactor model to llustrate certan smlartes. Summarzng (6.13) and (6.) provdes the followng representaton of the stochastc component of the return of the jth prmary securty account the volatlty 0 k R +1 dx k=1 0 k p ; j k (k) Z +1 (6.14) for f0 1 ::: n; 1g and j f0 1 ::: dg. Smlarly, usng (6.11), (6.1) and (6.), we obtan for the stochastc part of the return of the BBI the approxmate expresson R () +1 = H () +1 ; 1 log H () +1 ;X (0) +1 = dx k=1 p 0 k (k) Z +1 (6.15) for f0 1 ::: n; 1g. By (6.15), (6.14), (6.3), (6.4) and (6.5) we obtan dx (6.16) E R () +1 E R +1 dx k=1 k=1 0 k 0 k ; j k (6.17) 10
11 and E R +1 R() +1 dx k=1 0 k ; j k 0 k (6.18) for f0 1 ::: n; 1g. We can dene the followng approxmate rato of covarances of returns as generalzed jth beta factor at tme t, where P E R d +1 R() +1 k=1 = 0 k ; j k P 0 k d k=1 P d =1; l=1 0 k j k P 0 l d`=1 0 ` E R () +1 for f0 1 ::: n; 1g and j f0 1 ::: dg. The above relaton provdes the equvalent of the common beta factor n a benchmark framework. Note that we get for the domestc savngs account the beta factor (0) =0,ass to be expected. Moreover, for the generalzed beta factor () of a portfolo S () wth E R () () +1 R() +1 = E R () +1 one can show by smlar arguments as above that () =1; P d k=1 0 k P d j=0 j k P d`=1 0 ` : (6.19) Note that the generalzed beta factor of the semparametrc benchmark model s tme dependent and matches the wellknown return relatonshp of the Captal Asset Prcng Model, see Merton (1973). Fnally, for the GOP S () t can be shown, see Platen (001a), that dx j=0 j k 0 (6.0) for f0 1 ::: n; 1g and k f1 ::: dg. Therefore, usng (6.19) and (6.0) the generalzed beta factor () of the GOP, and thus abbi,s approxmately one. 7 Generalzed Hyperbolc Benchmark Models For the class of semparametrc benchmark models one can now specfy approprate famles of dstrbutons for the sources of uncertanty Z (1) ::: Z (d). 11
12 Let us choose the logreturn dstrbuton from the well establshed class of generalzed hyperbolc dstrbutons. These dstrbutons were extensvely studed n BarndorNelsen (1978) and BarndorNelsen & Blaesld (1981). In the followng statstcal analyss we assume, for smplcty, that the centralzed logreturns of benchmarked share prces have a symmetrc generalzed hyperbolc dstrbuton. We acknowledge that a slght skewness n logreturns s typcal. However, for the small tme step sze consdered here, ths s a hgher order eect. The man feature that we explore n the followng concerns the shape of the tals of the probablty densty f X of centralzed logreturns X. The symmetrc generalzed hyperbolc densty s of the form f X (x) = 1 p K () K ; 1 r r (x ; ) (x ; )! 1 (; 1 ) (7.1) for x <, where <, 0 and = wth 6= 0 f 0 and 6= 0f 0. Here K () s the moded Bessel functon of the thrd knd wth ndex. For smplcty, we assume that the above parameters reman constant over tme. Note that the symmetrc generalzed hyperbolc dstrbuton s a four parameter famly of dstrbutons. The two shape parameters are and, dened so that they are nvarant under scale transformaton as descrbed below. The kurtoss of the logreturn X s gven by the expresson X = 3 K () K + () K +1 () (7.) for ( ) [0 1) <. We remark that for =0and [; 0] we have nnte kurtoss. The parameter n (7.1) s a locaton parameter, where the logreturn X has mean m X =. We dene the parameter c as the unque scale parameter such that the varance of X s v X = c and ( f =0.e., >0 =0 c = K +1() K () otherwse: It can be shown that for!1or!1the symmetrc generalzed hyperbolc dstrbuton asymptotcally approaches the Gaussan dstrbuton. Thus the logreturns of the lognormal model, see Black & Scholes (1973), appear as lmtng cases of the above class of symmetrc generalzed hyperbolc logreturn dstrbutons. 1
13 We wll now descrbe four partcularly mportant symmetrc generalzed hyperbolc dstrbutons that concde wth the logreturn dstrbutons of mportant asset prce models that have been suggested by derent authors: The Student t dstrbuton was orgnally suggested by Praetz (197) as a sutable dstrbuton for asset returns. The Student t dstrbuton s a specal symmetrc generalzed hyperbolc dstrbuton, where one sets = 0 and <0. The parameter = ; (7.3) s called the degrees of freedom of the Student t dstrbuton. We have nte varance for > and nte kurtoss r =3 ; for >4. The Student t dstrbuton s a three parameter dstrbuton where smaller degrees of freedom ;4 mply heaver tals. BarndorNelsen (1995) proposed a model, where the logreturns generate a normalnverse Gaussan mxture dstrbuton. Ths dstrbuton appears as a specal case of the symmetrc generalzed hyperbolc dstrbuton where the shape parameter s xed at the level = ;0:5. The normalnverse Gaussan dstrbuton s also a three parameter dstrbuton. A smaller shape parameter mples larger tal heavness wth kurtoss r =3(1+ ;1 ): Eberlen & Keller (1995) and also Kuchler et al. (1995) suggested asset prce models, where logreturns are hyperbolcly dstrbuted. The hyperbolc dstrbuton results as a specal case of the symmetrc generalzed hyperbolc dstrbuton f we choose the shape parameter = 1, see Eberlen (001) for a recent survey. The hyperbolc dstrbuton s also a three parameter dstrbuton. Its kurtoss r = 3 K 1() K 3 () K () depends on the shape parameter, whch can be shown to reach ts maxmum value of r = 6 for =0. Madan & Seneta (1990), see also Geman, Madan & Yor (1998), proposed a class of models that result n logreturns whch follow a varance gamma dstrbuton. Ths dstrbuton s the specal case of the symmetrc generalzed hyperbolc dstrbuton that arses f one sets = 0 and the shape parameter strctly postve, that s > 0. A smaller mples heaver tals for ths three parameter dstrbuton wth kurtoss r = 3 (1 + ;1 ). For the lmtng case = 0 we have nnte kurtoss. 8 Testng Benchmark Models In ths secton we dentfy a dstrbuton that best ts the growth ratos of BBIs and benchmarked stock prces. We note that for several of the 13
14 aforementoned dstrbutons the kurtoss can be nnte. Therefore, asta tstcal method that reles on hgher order moments should not be used. The maxmum lkelhood approach avods ths problem. We recall, for xed j f1 ::: dg the logreturns X, f0 1 ::: n; 1g, are ndependent and dentcally dstrbuted. We dene the lkelhood rato n the standard form = Lm L gm, where L m represents the maxmzed lkelhood functon of a partcular three parameter dstrbuton, for nstance, the Student t dstrbuton. Wth respect to ths dstrbuton the maxmum lkelhood estmate for the parameters s computed. On the other hand, L gm denotes the maxmzed lkelhood functon for the four parameter symmetrc generalzed hyperbolc dstrbuton. As n! 1 the test statstc L n = ;ln s asymptotcally chsquare dstrbuted, see Rao (1973), wth degrees of freedom one. Asymptotcally, we then have the relaton ; ; P L n <1; 1 F (1) 1; 1 =1; (8.1) as n!1, where F (1) denotes the chsquare dstrbuton wth one degree of freedom and 1; 1 ts 100(1 ; )% quantle. We can then check, say, for the standard 99% sgncance level whether or not the test statstc L n s n the 1% quantleofthechsquare dstrbuton wth one degree of freedom. Ths means, f the relaton L n < 0:01 1 0:000 (8.) s not satsed, then we reject on a 99% sgncance level the hypothess that the suggested dstrbuton s the true underlyng logreturn dstrbuton. We study benchmarked US stock prce logreturns on the bass of daly benchmarked share prce data, provded by Thomson Fnancal for the eleven year perod from 1987 to 1997, usng the S&P500 as BBI. The total number of observed daly logreturns for each benchmarked stock s about 500, whch provdes relable asymptotc results for the maxmum lkelhood rato test. In Table 1 weshow the test statstcs L n for twenty leadng shares of the US market. The stock codes correspond to those commonly used. The smallest L n value dentes the best t and s dsplayed n bold type. We note that for the majorty of logreturns of benchmarked stock prces the Student t dstrbuton gves the smallest L n value and thus yelds the best t n the class of symmetrc generalzed hyperbolc dstrbutons consdered. We observe that fourteen of the twenty L n values n Table 1 appear to be smaller than the 0:01 1 =0:000 quantle. For these benchmarked stock prce logreturns the hypothess that the Student t dstrbuton s the true underlyng dstrbuton cannot be rejected at a 99% sgncance level. A smlar study has been performed for the Australan, German, Japanese and UK market usng the correspondng market ndex as BBI. Detaled results for the other four markets can be obtaned from the authors. For thrty one 14
15 Inverse Varance Stock Student t Gaussan Hyperbolc Gamma GE KO XON INTC MRK RD IBM MO PG PFE BMY T WMT JNJ LLY DD DIS HWP PEP MOB Table 1: The L n values for logreturns of benchmarked US stocks. of the one hundred stocks consdered, the Student t dstrbuton could not be rejected at a 99% sgncance level as the true underlyng dstrbuton. In all ve markets the Student t dstrbuton clearly provdes the best t for most logreturns of the benchmarked stocks. To llustrate the results for all ve markets we plot n Fgure 1 the estmated ( )parameter values for the one hundred examned benchmarked stocks. These estmates characterze the specc shape of the tals for the estmated symmetrc generalzed hyperbolc denstes. The postve partof the axs n Fgure 1 parameterzes the varance gamma dstrbuton. We note that only one of the one hundred stocks generated an ( )estmate on the postve axs. Consequently, thevarance gamma dstrbuton does not t our data well. The hyperbolc dstrbuton s represented by pars of shape parameters ( ) wth =1. There are about three to four ponts n Fgure 1 that are located n the neghborhood of the horzontal lne =1. Thus the t of the hyperbolc dstrbuton s also qute lmted. The ponts ( ) on the horzontal lne = ; 1 correspond to the normalnverse Gaussan dstrbuton. We note that several stocks generate ponts n the area close to ths lne. Therefore, the normalnverse Gaussan dstrbuton provdes a 15
16 reasonable t to the data. We remark that for small the normalnverse Gaussan dstrbuton concdes asymptotcally wth the Student t dstrbuton and most parameter estmates are less than one. It s obvous that most of the one hundred ( )estmates are concentrated close to the negatve axs. Ths s the area that s characterstc for dstrbutons that are smlar to the Student t dstrbuton. It appears that the ( )estmates support a Student t dstrbuton wth a parameter value for of approxmately ;. Ths corresponds to a Student t dstrbuton of four degrees of freedom, see (7.3). The average estmated degrees of freedom for the Student t dstrbuton obtaned from all benchmarked stocks was ^ =4: lambda alphabar Fgure 1: ( )plot for logreturns of benchmarked stocks. In a study smlar to that descrbed above t has been shown n Hurst & Platen (1997) that the logreturns of the BBIs of the equty markets of most developed economes, ncludng the ve also consdered here, support the Student t dstrbuton, agan wth degrees of freedom close to four. As an alternatve to the above mentoned studes, a seres of papers, see, for nstance, Dacarogna et al. (1995), has shown that the tals of asset logreturns follow approxmately a power law, where the estmates of the socalled tal ndex are typcally n the range of three to ve. Ths s consstent wth our ndngs for benchmarked stock logreturns. Furthermore, we remark that the Student t benchmark model can be nterpreted as a dscrete tme analog of the, socalled, mnmal market model, see Platen (001a, 001b), whch models the dynamcs of a GOP that exhbts mnmum varance n ts drft. These lead to Student t dstrbuted logreturns of the GOP wth exactly = 4 degrees of freedom. 16
17 9 VaR Analyss As outlned n the ntroducton and n Secton 3, the modelng of event rsk n nternal models s of ncreasng mportance n VaR analyss. The above class of semparametrc benchmark models s able to encompass event rskby choosng an adequate leptokurtc dstrbuton for the sources of uncertanty. As prevously explaned, the Student t dstrbuton provdes an excellenttto logreturns of benchmarked equty data. Incorporatng event rsk completes VaR modelng for measurng market rsk by takng nto account all regulatory subcategores of market rsk, that s general, dosyncratc and event rsk. As an alternatve to the above approach, Gbson (001) used a mxng dstrbuton to specfy a ve parameter model that puts sucent mass nto the tals of logreturn dstrbutons by ntroducng derent regmes for the means of certan mxed dstrbutons. Due& Pan (001) apply jumpdusons for modelng specc and event rsk n the context of VaR whch reles on varous parameters. Though the class of jumpdusons s ntutvely appealng, the parameters are dcult to estmate, see Honore (1998). Huschens & Km (1999) studed a VaR model that uses Student t dstrbuted returns but not n a benchmark settng. Note that we specfy n a parsmonous way the sources of uncertanty n the followng verson of the Student tbenchmark model. We explot the fact that symmetrc generalzed hyperbolc dstrbutons admt a representaton of a mxture of normal dstrbutons. Ths means, f one chooses the varance of a condtonally Gaussan dstrbuton as the nverse of a Gamma dstrbuted random varable, then the resultng dstrbuton s a Student t dstrbuton. To generate the Student t dstrbuted sources of uncertanty Z (0) +1, :::, Z(d) +1 approprately, weset where +1 denotes the market actvty wth +1 = 1 ; 1 X Z (k) +1 = p +1 Y (k) +1 (9.1) `=1 (`) +1! ;1 (9.) for f0 1 ::: n; 1g and k f1 ::: dg wth f3 4 :::g. Addtonally to the ndependent standard Gaussan random varables Y (k) +1 that appear n (9.1) we employ further ndependent standard Gaussan random varables (`) +1. Hence, the random varables +1 are chsquare dstrbuted wth degrees of freedom. Then the random varables Z (k) +1 are Student t dstrbuted wth unt varance and degrees of freedom. The market actvty +1 can be nterpreted as a daly ncrement of the random busness tme of the market. Note, the market actvty converges to 1 as the degrees of freedom tend to nnty, whch yelds the lognormal benchmark model. 17
18 In addton to the typcal parameters of the lognormal benchmark model we have used here only the extra parameter, whch s sucent tocharac terze the leptokurtoss of the Student t dstrbuton. As shown prevously, the typcal parameter choce for s about 4. Smaller degrees of freedom generate logreturns wth more extreme movements. An mportant feature of the resultng multvarate Student t dstrbuton s ts copula. It realstcally captures the dependence of extreme asset prce movements as shown n Embrechts, McNeal & Straumann (1999) and related work. In our model the Student tcopula, characterzng the jont occurrence of extreme moves n logreturns of several benchmarked stocks, s automatcally obtaned. To calbrate the above benchmark Student t model one needs to estmate the volatltes j k,typcally obtaned from a standard lognormal calbraton. The above model then represents a smple extenson of the lognormal model, where one needs only to add an estmate for the degrees of freedom. It s reasonable to set =4wthout losng much accuracy, as we wll see from Table. The equatons (9.1) and (9.) gve aprescrpton that can also be used for smulaton purposes. They nvolve the Cholesky decomposton D of the covarance matrx, see (6.9). The Student t benchmark model provdes qute accurate results f the VaR calculaton s based on extensve Monte Carlo smulatons. However, reasonably accurate MonteCarlo smulatons are extremely tme consumng. To crcumvent ths problem, n the followng we propose a hghly ecent method for VaR calculatons. In practce, equty portfolos, are typcally domnated by large lnear portfolos. Our model s n that case analytcally tractable. Note that the jont ::: X(d) +1 )> s a multvar dstrbuton of the random vector X +1 =(X (1) +1 ate Student t dstrbuton wth degrees of freedom, where Snce Y +1 s Gaussan and 1 +1 X +1 = p +1 D Y +1 : s ndependent chsquare dstrbuted, the resultng multvarate Student t dstrbuton for X +1 belongs to the class of ellptcal dstrbutons. For lnear portfolos the calculaton of VaR numbers s for ths class of dstrbutons analytcally tractable. More precsely, a theorem n Fang, Kotz & Ng (1990) yelds the representaton a > X +1 = ka > D k +1 (9.3) for any gven weght vector a, where kk s the Eucldean norm, a < d, D > D = and +1 denotes a Student t dstrbuted scalar random varable wth degrees of freedom. The representaton (9.3) sgncantly smples the VaR calculaton even for lnear portfolos wth an extremely large number of consttuents. Furthermore, we menton that f a portfolo wth many consttuents s not lnear but represents a dversed portfolo n the sense 18
19 descrbed by Platen (00), then t approxmates the GOP and thus also our benchmark, the BBI. These asymptotcs yeld accurate VaR numbers and save computatonal tme when compared to standard MonteCarlo smulaton. Snce the multvarate Student t dstrbuton s an ellptcal dstrbuton, t s shown by Embrechts, McNeal & Straumann (1999), that VaR s n ths case a coherent rsk measure, see Artzner et al. (1997). Thus the crtcal subaddtvty ofvar for the Student tbenchmark model s guaranteed. Ths fact s hghly mportant for the consstent use of VaR as an nternal rsk measure. Snce t allows the nternal captal allocaton to partcular busness lnes, the Student t benchmark model oers sgncant support for nternal rsk management. In order to calculate VaR for short term horzons we apply the socalled square root tme rule. Ths approxmaton s n lne wth the regulatory recommendatons of the Basle Commttee. From (9.3) we obtan then the followng formula for the VaR number of a gven portfolo S () at tme t. VaR h (S () ) V pa > a p h ~ t (): (9.4) Here V denotes the market value of the portfolo at tme t, p a > a characterzes the volatlty of the portfolo, the tme step sze for a tradng day, h the number of tradng days and ~ t () thequantle of the standardzed Student t dstrbuton wth degrees of freedom. Obvously, the product (9.4) generalzes the wellknown short hand formula, used n RskMetrcs, to calculate VaR when one uses the event factor that s ' = ~ t () q (9.5) VaR h (S () ) V pa > aq a p h ' (9.6) where q s the quantle of the standard Gaussan dstrbuton. Consequently, the event factor' adjusts the standard VaR formula to a level that captures approxmately event rsk when one uses the Student t benchmark model as nternal model. Accordng to the quantles of the Gaussan and the Student t dstrbuton one obtans by (9.4) the event factors shown n Table ' Table : Event factor ' n dependence on degrees of freedom. Even for rather small degrees of freedom, say, the addtonal regulatory captal wll not surpass 16%. To conrm the approprate choce of the model 19
20 and ts calbraton one needs to perform stress tests. Along these lnes, Gbson (001) performed an extensve study usng representatve portfolos for US nsttutons, whch show that the event factor should be close to 1:1. In other words, by means of extensve hstorcal smulaton Gbson (001) dented an event factor of ' =1:1. Ths s the event factor that exactly matches n Table the degrees of freedom =4,whch strongly supports our emprcal ndng of approxmately four degrees of freedom for logreturns. References Alexander, C. (1996). Handbook of Rsk Management and Analyss. Wley, Chchester. Artzner, P., F. Delbaen, J. M. Eber, & D. Heath (1997). Thnkng coherently. Rsk 10, 68{71. BarndorNelsen, O. (1978). Hyperbolc dstrbutons and dstrbutons on hyperbolae. Scandnavan Journal of Statstcs 5, 151{157. BarndorNelsen, O. (1995). NormalnnInverse Gaussan processes and the modellng of stock returns. Techncal report, Unversty of Aarhus BarndorNelsen, O. & P. Blaesld (1981). Hyperbolc dstrbutons and ramcatons: Contrbutons to theory and applcaton. In C. Talle, G. P. Patl, and B. A. Baldessar (Eds.), Statstcal Dstrbutons n Scentc Work, Volume 4, pp. 19{44. Dordrecht: Redel. Basle (1996a). Amendment to the Captal Accord to Incorporate Market Rsks. Basle Commttee on Bankng and Supervson, Basle, Swtzerland. Basle (1996b). Modcatons to the Market Rsk Amendment. Basle Commttee on Bankng and Supervson, Basle, Swtzerland. Black, F. & M. Scholes (1973). The prcng of optons and corporate labltes. J. Poltcal Economy 81, 637{659. Dacarogna, M., U. A. Muller, O. V. Pctet, & C. G. De Vres (1995). The dstrbuton of extremal foregn exchange rate returns n extremely large data sets. A preprnt of the Olsen Group, Zurch. Due, D. &J.Pan (1997). An overvew of Value at Rsk. J. of Dervatves 4(3), 7{49. Due, D. & J. Pan (001). Analytc Value at Rsk wth jumps and credt rsk. Fnance and Stochastcs 5, 155{180. Eberlen, E. (001). Applcatons of generalzed Levy motons n nance. In O. E. BarndorNelsen, T. Mkosch, and S. Resnck (Eds.), Levy Processes: Theory and Applcaton. Eberlen, E. & U. Keller (1995). Hyperbolc dstrbutons n nance. Bernoull 1, 81{99. 0
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